I've cancelled by NYTimes subscription just a few months ago. Didn't have to do any phone calls-- just answered a few questions in the chat, right there on their website.
Instead I simply sent them an email insisting that I wanted to cancel, and that I refused to talk to them on the phone to do it. They replied with a “great offer” to reduce the price. In my reply I simply insisted on cancelling. I didn’t hear from them again, except for a confirmation by post a while later that the service had been cancelled.
Of course, this is much more effort than it should be, especially an emotional effort because being treated like this is very upsetting. But it might be worth trying.
there is no "just" about that...